Physics of the Piano Piano Tuners Guild, June 5, 2000

1. Physics of the PianoPiano Tuners Guild, June 5, 2000 Charles E. Hyde-Wright, Ph.D. Associate Professor of Physics Old Dominion University Norfolk VA chyde@odu.edu Piano Tuners Guild

2. The Physics of Music and Musical Reproduction:Autumn 2000, Mon.&Wed. 4:20-5:35pm Call # 18539 ODU PHYS 332W Topics: Physical attributes of music and sound. Acoustics of musical instruments and concert halls. Electronic generation and recording of sound. Neuro-physiology of sound perception. Piano Tuners Guild

3. Physics of the Piano • Oscillations & Sound • Vibrations of a String • Travelling waves & Reflections • Standing Waves • Harmonics • Piano acoustics • Hammer action • Sound Board • Multiple Strings • Chords, Scales & Tuning Piano Tuners Guild

4. Why does a mass on a spring oscillate? • It is not because I push it • The mass continues long after I let go. • The spring is pushing on the mass. • Why doesn’t the mass just come to rest in the middle? • After all, the spring(s) exert no (net) force on the mass when it is exactly in the middle. • No force seems like no motion (wrong). Piano Tuners Guild

5. Force and Motion • The mass moves even when nothing is pushing • The mass moves because of inertia • Forces do not cause motion…forces cause motion to change • Force is proportional (mass) to the time rate of change of motion (acceleration) • F = ma • A force acting to left either: • Makes the mass go faster to left; or • Makes the mass slow down as it moves to right Piano Tuners Guild

6. [Net] Force causes motion to change • F=ma • The time rate of change of velocity (acceleration) is proportional (mass) to the net applied force. • A force acting to left either: • Makes the mass go faster to left; or • Makes the mass slow down as it moves to right Piano Tuners Guild

7. For a Spring, F  -kx Force pushes towards middle Force grows with distance Force = -kx and Force = ma Acceleration = a = - (k/m)x Frequency (f) measures how fast an oscillation is changing Acceleration is rate of change of rate of change of position Car on freeway on-ramp: 60 miles per hour per 10 sec Acceleration is equal to a mathematical constant times frequency squared times position. F=ma & Mass on a Spring • Frequency increases with stiffness, decreases with mass Piano Tuners Guild

8. Vibrations of a String • Each little segment of a string is like a mass on a spring • The spring force is supplied by the tension in the string and the curvature of the wave. • A wave (of arbitrary shape) travels on a string with velocity Piano Tuners Guild

9. Travelling waves and Reflections • Each end of the string is held rigidly. • To the wave, the fixed point acts like a wave of opposite amplitude travelling in opposite direction. • Rigid end of string reflects wave with opposite sign • Loose end of string (or other wave--e.g. organ pipe) reflects wave with equal sign. Piano Tuners Guild

10. Standing Waves • Each point on string experiences waves reflecting from both ends of string. • For a repeating wave (e.g. sinusoidal) • Velocity = wavelength times frequency: v = l f • The superposition of reflecting waves creates a standing wave pattern, but only for wavelengths l = 2L, L, L/2, … = 2L/n) • Only allowed frequencies are f = n v/(2L) • Pitch increases with Tension, decreases with mass or length Piano Tuners Guild

11. Harmonics on string • Plot shows fundamental and next three harmonics. • Dark purple is a weighted sum of all four curves. • This is wave created by strumming, bowing, hitting at position L/4. • Plucking at L/2 would only excite f1, f3, f5, ... Piano Tuners Guild

12. Pitch, Timbre, & Loudness • Equal musical intervals of pitch correspond to equal ratios of frequency: • Two notes separated by a perfect fifth have a frequency ratio of 3:2. • Notice that 2nd and 3rd harmonic on string are perfect 5th • Timbre is largely determined by content of harmonics. • Clarinet, guitar, piano, human voice have different harmonic content for same pitch • Loudness is usually measured on logarithmic decibel (tenths of bel) scale, relative to some arbitrary reference intensity. • 10 dB is a change in sound intensity of a factor of 10 • 20 db is a change in sound intensity of a factor of 100. Piano Tuners Guild

13. Frequency analysis of sound • The human ear and auditory cortex is an extremely sophisticated system for the analysis of pitch, timbre, and loudness. • My computer is not too bad either. • Microphone converts sound pressure wave into an electrical signal. • Computer samples electrical signal 44,000 times per sec. • The stream of numbers can be plotted as wave vs. time. • Any segment of the wave can be analysed to extract the amplitude for each sinusoidal wave component. Piano Tuners Guild

14. Samples of Sound Sampling • Clarinet • Guitar • Piano • Human Voice • ... Piano Tuners Guild

15. Piano keys(Grand Piano) • Key is pressed down, • the damper is raised • The hammer is thrown against string • The rebounding hammer is caught by the Back Check. Piano Tuners Guild

16. Hammer action • Throwing the hammer against the string allows the hammer to exert a very large force in a short time. • The force of the hammer blow is very sensitive to how your finger strikes the key, but the hammer does not linger on the string (and muffle it). • From pianissimo (pp) to fortissimo (ff) hammer velocity changes by almost a factor of 100. • Hammer contact time with strings shortens from 4ms at pp to < 2 ms at ff (for middle C-264 Hz) • Note that 2 ms = ½ period of 264 Hz oscillation Piano Tuners Guild

17. From Strings to Sound • A vibrating string has a very poor coupling to the air. To move a lot of air, the vibrations of the string must be transmitted to the sound board, via the bridge. • The somewhat irregular shape, and the off center placement of the bridge, help to ensure that the soundboard will vibrate strongly at all frequencies • Most of the mystery of violin making lies in the soundboard. Piano Tuners Guild

18. Piano frame • A unique feature of the piano, compared to violin, harpsichord. is the very high tension in the strings. • This increases the stored energy of vibration, and therefore the dynamic power and range of the piano. • Over 200 strings for 88 notes,each at  200 lb tension • Total tension on frame > 20 tons. • The Piano is a modern instrument (1709, B. Cristofori): • High grade steel frame. • Also complicated mechanical action. Piano Tuners Guild

19. Piano strings • An ideal string (zero radius) will vibrate at harmonics • fn = n f1 • A real string (finite radius r) will vibrate at harmonics that are slightly stretched: • fn = n f1[1+(n2-1)r4k/(TL2)] • Small radius-r, strong wire (k), high tension (T), and long strings (L) give small in-harmonicity. • For low pitch, strings are wrapped, to keep r small Piano Tuners Guild

20. In-harmonicity & tone color • Perfect harmonics are not achievable--and not desirable. A little in-harmonicity gives richness to the tone, and masks slight detunings of different notes in a chord. • Each octave is tuned to the 2nd harmonic of the octave below. Piano Tuners Guild

21. Multiple Strings • Multiple Strings store more energy--louder sound • Strings perfectly in tune: • Sound is loud, but decays rapidly • Strings strongly out of tune: • Ugly beats occur as vibrations from adjacent strings first add, then cancel, then add again. • If strings are slightly out of tune • Sound decays slowly • Beats are slow, add richness to tone. Piano Tuners Guild

22. Multiple Strings, Power and Decay Time • Decay time of vibration = Energy stored in string divided by power delivered to sound board. • Power delivered to sound board = force of string * velocity of sound board (in response to force) • Three strings store 3 times the kinetic energy of one string • If three strings are perfectly in tune, Force is 3 times larger, velocity is three times larger, power is 9 times larger, Decay time is 3/9 = 1/3 as long as one string alone (Una corda pedal). • If strings are slightly mistuned, motion is sometimes in phase, sometimes out of phase, average power of three strings is only 3 times greater than power of one string. Decay time of 3 strings is SAME as decay time of one string alone—just louder. Piano Tuners Guild

23. Beats from mistuned strings • Two tones are mistuned by 10%. One string makes 10 oscillations in the time it takes the other to make 11 oscillations. • Yellow curve = resulting superposition of two waves • ½ of beat period is shown. Beat period = 20*period of individual wave. • Acoustic power would be 4x individual wave, if strings were perfectly in tune. Because of beats, average acoustic power is 2x individual contribution Piano Tuners Guild

24. Chords, Scales, and Tuning • For the simple chords of perfect fifths and thirds, the harmonics of each note match the fundamental of the other notes in the chord. • Equal tempered tuning was designed (by J.S. Bach & others) to give the closest possible match between pitch and harmonics within chords, for any possible starting note. Piano Tuners Guild

25. Conclusions • The mathematical theory of music dates to antiquity. • Music illustrates fundamental physical and mathematical concepts. • Music is influenced by technology • ‘Modern’ examples: • Piano, Clarinet • Musical Reproduction and Processing (Digital predates Analog!). • Music appreciation has many layers. • Science can add another layer Piano Tuners Guild